G.E. Andrews writes, "Leibniz was apparently the first person to ask about partitions. In a 1674 letter he asked J. Bernoulli about the number of “divulsions” of integers. In modern terminology, he was asking the first question about the number of partitions of integers. He observed that there are three partitions of 3 (3, 2 + 1, and 1 + 1 + 1) as well as five of 4 (4, 3 + 1, 2 + 2, 2 + 1 + 1 and 1 + 1 + 1 + 1)."

I find this assertion of priority incredibly difficult to believe as well as the following:

Peter Luschny, an editor for the OEIS commenting on simply an allusion to a listing of integer partitions, certainly not an attribution as to the first publication with that order, "I strongly appeal against the terms A&S-order or A-St-order. Whoever (...?) introduced these terms into the OEIS made a disservice. It is well known that this widely used order was introduced by C. F. Hindenburg in his 1779 dissertation and the proper way to reference it is either 'Hindenburg order' or a mathematical description in general terms. Neither Abramowitz nor Stegun have ever written a single line about monomial orders, not even in their Handbook. A detailed analysis of the Hindenburg way to enumerate the partitions can be found Knuth's chapter 7 in TAOCP 4. The term A&S does not appear a single time in this work. Let us base our terminology on serious literature rather than introduce private jargon into OEIS."

I would assume that the ancient Babylonians and Chinese had addresssed the topic long before either of these two mathematicians (in diverse, reasonable, and exhaustive orderings), and that this OCD necessitating attributing priority to particular, long-deceased individuals is both gratuitous and naive.

Accordingly, to support a definitive rebuttal of these somewhat trivial assertions and much more importantly to discourage such scholasticism in general, can anyone provide references on earlier investigations of the integer partitions?

(Some indication of interests of the ancients in integer partitions: The oldest known magic square dates to 2200 BCE (Cf: A brief survey of combinatorics).

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    $\begingroup$ Earlier than what ? After Leibniz, we have Ch.XVI DE PARTITIONE NUMERORUM of Leonhard Euler, Introductio in analysin infinitorum (1748). $\endgroup$ Commented Oct 7, 2016 at 7:33
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    $\begingroup$ See Brian Hopkins a and Robin Wilson, Euler’s Science of Combinations for modern comment. $\endgroup$ Commented Oct 7, 2016 at 7:36
  • $\begingroup$ I am not aware of combinatorial problems in ancient Babylonian, Egyptian or Chinese sources (I Ching and Lo Shu can be related to it today, but there is no sign of that in contemporary sources), and of only one in the entire Greek corpus. Even permutations and combinations first appear in India only in 6th century AD, and it is a step up in abstraction from counting collections of objects to counting ways to partition. Probability does not appear before 17th century despite prolific gambling in antiquity. So I am inclined to credit Andrews's claim. $\endgroup$
    – Conifold
    Commented Oct 7, 2016 at 21:15
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    $\begingroup$ Nārāyaṇa Paṇḍita in his Gaṇita-kaumudī (c. 1356) enumerates ordered q-partitions of an integer, i.e. the number of ways of writing an integer n as an ordered sum of parts all ≤ q. (He was generalizing from an earlier (centuries-old) enumeration of the number of ways of writing n as a sum of 1s and 2s, which gives the Fibonacci numbers.) However I'm not aware of him enumerating unordered partitions. $\endgroup$ Commented Oct 7, 2016 at 23:56
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1 Answer 1


Leibniz published in a book regarding this problem prior any of those letters.

Leibnuetzio, Gottfredo, 1666, Dissertatio De Arte Combinatoria, Lipsiae: Ficki, Seubold, p. 59. (Problem 3)

The fact that this is listed as a problem he is trying to solve suggests the problem is much older. Which it probably is ... because of gambling. Likely this is one of the earlier publications however.

Euler published (as M.A. pointed out) his book somewhat later but gave a general formula and the modern statement (and name) of the problem.

Eulero, Leonhardo, 1748, Introductio In Analysin Infinitorum, I, Lausannae: Bousquet, p. 255.


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